A chemist wants to create a solution that is 24% acid. How many liters of a 40% acid solution must be added to 50 liters of an 18% acid solution to obtain this 24% acid mixture?
.40x + .18*50 = .24(x+50)
To solve this problem, we need to understand the concept of mixing different solutions with different concentrations to obtain a desired concentration.
Let's break down the problem step-by-step:
1. Identify the given information:
- We have 50 liters of an 18% acid solution.
- We want to add a 40% acid solution.
- The desired mixture should be a 24% acid solution.
2. Assign variables to the unknowns:
- Let's call the volume of the 40% acid solution we need to add as "x" liters.
3. Set up equations using the principle of concentration:
- The acid content is given as a percentage, so we can use the equation:
Acid content = (Volume of acid / Total volume) * 100
- For the initial 18% acid solution, the equation becomes:
18 = (Volume of acid in 18% solution / 50) * 100
- For the final 24% acid mixture, the equation becomes:
24 = [(Volume of acid in 18% solution + Volume of acid in 40% solution) / (50 + x)] * 100
4. Solve the equations:
- From the first equation, we can calculate the volume of acid in the 18% solution:
Volume of acid in 18% solution = (18/100) * 50 = 9 liters
- Substituting the known values into the second equation:
24 = [(9 + Volume of acid in 40% solution) / (50 + x)] * 100
- Simplify the equation:
24 = (9 + Volume of acid in 40% solution) / (50 + x)
- Cross-multiply and solve for the unknown:
24(50 + x) = 9 + Volume of acid in 40% solution
1200 + 24x = 9 + Volume of acid in 40% solution
Volume of acid in 40% solution = 24x + 1191
5. Set up an equation to relate the volume of acid in the 40% solution to its concentration:
- Since the desired mixture is 24% acid, we can write:
40% of x liters + 9 liters (from the initial solution) = 24% of (50 + x) liters
- Converting the percentages to decimals:
0.4x + 9 = 0.24(50 + x)
6. Solve for x:
- Distribute the multiplication on the right side:
0.4x + 9 = 12 + 0.24x
- Simplify and isolate x:
0.16x = 3
x = 3 / 0.16
x ≈ 18.75
Based on the calculations, the chemist would need to add approximately 18.75 liters of the 40% acid solution to the 50 liters of the 18% acid solution to obtain a 24% acid mixture.